Time-dependent moments of the counts on a BMAP

Consider a batch Markovian arrival process (BMAP) as a counting process over an underlying Markov process representing the state of environment. Such a process is useful as a model of correlated inputs, that is, burst traffics made of video and voice, for example. We consider Laplace transformation of the first and second factorial moments of the counts of the BMAP. From this, we get the eigenvalue expression for these moments without assuming distinct eigenvalues of the infinitesimal generator. In this formula, matrix exponential functions are replaced by ordinary exponentials, and the exact time-dependent form of the moments are also obtained. This seems to be profitable for model fitting.